Locally seeded embeddings, and Ramsey numbers of bipartite graphs with sublinear bandwidth
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913563368488960 |
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| author | Altschuler, Dylan J. Huang, Han Tikhomirov, Konstantin |
| author_facet | Altschuler, Dylan J. Huang, Han Tikhomirov, Konstantin |
| contents | A seminal result of Lee asserts that the Ramsey number of any bipartite $d$-degenerate graph $H$ satisfies $\log r(H) = \log n + O(d)$. In particular, this bound applies to every bipartite graph of maximal degree $Δ$. It remains a compelling challenge to identify conditions that guarantee that an $n$-vertex graph $H$ has Ramsey number linear in $n$, independently of $Δ$. Our contribution is a characterization of bipartite graphs with linear-size Ramsey numbers in terms of graph bandwidth, a notion of local connectivity. We prove that for any $n$-vertex bipartite graph $H$ with maximal degree at most $Δ$ and bandwidth $b(H)$ at most $\exp(-CΔ\logΔ)\,n$, we have $\log r(H) = \log n + O(1)$. This characterization is nearly optimal: for every $Δ$ there exists an $n$-vertex bipartite graph $H$ of degree at most $Δ$ and $b(H) \leq \exp(-cΔ)\,n$, such that $\log r(H) = \log n + Ω(Δ)$. We also provide bounds interpolating between these two bandwidth regimes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18223 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Locally seeded embeddings, and Ramsey numbers of bipartite graphs with sublinear bandwidth Altschuler, Dylan J. Huang, Han Tikhomirov, Konstantin Combinatorics Discrete Mathematics Data Structures and Algorithms 05D10 A seminal result of Lee asserts that the Ramsey number of any bipartite $d$-degenerate graph $H$ satisfies $\log r(H) = \log n + O(d)$. In particular, this bound applies to every bipartite graph of maximal degree $Δ$. It remains a compelling challenge to identify conditions that guarantee that an $n$-vertex graph $H$ has Ramsey number linear in $n$, independently of $Δ$. Our contribution is a characterization of bipartite graphs with linear-size Ramsey numbers in terms of graph bandwidth, a notion of local connectivity. We prove that for any $n$-vertex bipartite graph $H$ with maximal degree at most $Δ$ and bandwidth $b(H)$ at most $\exp(-CΔ\logΔ)\,n$, we have $\log r(H) = \log n + O(1)$. This characterization is nearly optimal: for every $Δ$ there exists an $n$-vertex bipartite graph $H$ of degree at most $Δ$ and $b(H) \leq \exp(-cΔ)\,n$, such that $\log r(H) = \log n + Ω(Δ)$. We also provide bounds interpolating between these two bandwidth regimes. |
| title | Locally seeded embeddings, and Ramsey numbers of bipartite graphs with sublinear bandwidth |
| topic | Combinatorics Discrete Mathematics Data Structures and Algorithms 05D10 |
| url | https://arxiv.org/abs/2410.18223 |